Potential Analysis ( IF 1.0 ) Pub Date : 2020-05-30 , DOI: 10.1007/s11118-020-09852-6 Jie-Xiang Zhu
We establish some exact asymptotic results for a matching problem with respect to a family of beta distributions. Let X1,…,Xn be independent random variables with common distribution the symmetric Jacobi measure \(d\mu (x) = C_{d} (1-x^{2})^{\frac d2 -1} dx\) with dimension d ≥ 1 on [− 1,1], and let \(\mu _{n} = \frac {1}{n} {\sum }_{i = 1}^{n} \delta _{X_{i}}\) be the associated empirical measure. We show that
$$ \lim\limits_{n \to \infty} n{\mathbb{E}} \left[ {W_{2}^{2}}(\mu^{n}, \mu ) \right] = \sum\limits_{k = 1}^{\infty} \frac{1}{k(k+d-1)}, $$where W2 is the quadratic Kantorovich distance with respect to the intrinsic cost \(\rho (x, y) = |\arccos (x) - \arccos (y)|\), (x,y) ∈ [− 1,1]2, associated to the model. When μ is the product of two Jacobi measures with dimensions d and \(d^{\prime }\) respectively, then
$$ {\mathbb{E}} \left[ {W_{2}^{2}}(\mu^{n}, \mu ) \right] \approx \frac{\log n}{n} . $$In the particular case \(d = d^{\prime } = 1\) (corresponding to the product of arcsine laws),
$$ \lim_{n \to \infty} \frac{n}{\log n} {\mathbb{E}} \left[ {W_{2}^{2}}(\mu^{n}, \mu ) \right] = \frac{\pi}{4}. $$Similar results do hold for non-symmetric Jacobi distributions. The proofs are based on the recent PDE and mass transportation approach developed by L. Ambrosio, F. Stra and D. Trevisan.
中文翻译：
Jacobi模型最优匹配问题的一些结果
我们针对一族beta分布为匹配问题建立了一些精确的渐近结果。令X 1，…，X n为具有共同分布的独立随机变量，对称雅可比测度\（d \ mu（x）= C_ {d}（1-x ^ {2}）^ {\ frac d2 -1} dx \）与尺寸d上≥1 [ - 1,1]，和让\（\亩_ {N} = \压裂{1} {N} {\总和} _ {i = 1} ^ {N} \增量_ {X_ {i}} \）是相关的经验指标。我们证明
$$ \ lim \ limits_ {n \ to \ infty} n {\ mathbb {E}} \ left [{W_ {2} ^ {2}}（\ mu ^ {n}，\ mu）\ right] = \ sum \ limits_ {k = 1} ^ {\ infty} \ frac {1} {k（k + d-1）}，$$其中W 2是相对于内在成本\（\ rho（x，y）= | \ arccos（x）-\ arccos（y）| \）的二次Kantorovich距离，（x，y）∈[− 1， 1] 2，与模型关联。当μ是两个Jacobi量度分别为d和\（d ^ {\ prime} \）的乘积时，则
$$ {\\ mathbb {E}} \ left [{W_ {2} ^ {2}}（\ mu ^ {n}，\ mu）\ right] \ approx \ frac {\ log n} {n}。$$在特定情况下\（d = d ^ {\ prime} = 1 \）（对应于反正弦定律的乘积），
$$ \ lim_ {n \ to \ infty} \ frac {n} {\ log n} {\ mathbb {E}} \ left [{W_ {2} ^ {2}}（\ mu ^ {n}，\ mu）\ right] = \ frac {\ pi} {4}。$$对于非对称的Jacobi分布也有类似的结果。证据基于L.Ambrosio，F.Stra和D.Trevisan最近开发的PDE和大众运输方法。
京公网安备 11010802027423号